Properties

Label 882.5.c.a.685.4
Level $882$
Weight $5$
Character 882.685
Analytic conductor $91.172$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 882.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(91.1723074400\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 685.4
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 882.685
Dual form 882.5.c.a.685.3

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843 q^{2} +8.00000 q^{4} +23.9515i q^{5} +22.6274 q^{8} +O(q^{10})\) \(q+2.82843 q^{2} +8.00000 q^{4} +23.9515i q^{5} +22.6274 q^{8} +67.7452i q^{10} -97.9706 q^{11} -104.211i q^{13} +64.0000 q^{16} +107.676i q^{17} +38.3935i q^{19} +191.612i q^{20} -277.103 q^{22} +1021.50 q^{23} +51.3238 q^{25} -294.755i q^{26} -621.603 q^{29} +1519.14i q^{31} +181.019 q^{32} +304.553i q^{34} -562.235 q^{37} +108.593i q^{38} +541.961i q^{40} +1023.20i q^{41} -3382.41 q^{43} -783.765 q^{44} +2889.23 q^{46} +3945.03i q^{47} +145.166 q^{50} -833.692i q^{52} -2190.60 q^{53} -2346.55i q^{55} -1758.16 q^{58} +2934.87i q^{59} +665.346i q^{61} +4296.76i q^{62} +512.000 q^{64} +2496.03 q^{65} -5925.38 q^{67} +861.405i q^{68} +4494.41 q^{71} -8968.93i q^{73} -1590.24 q^{74} +307.148i q^{76} -10446.8 q^{79} +1532.90i q^{80} +2894.04i q^{82} -1269.28i q^{83} -2579.00 q^{85} -9566.89 q^{86} -2216.82 q^{88} +3604.36i q^{89} +8171.99 q^{92} +11158.2i q^{94} -919.584 q^{95} -1950.53i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{4} + O(q^{10}) \) \( 4q + 32q^{4} - 324q^{11} + 256q^{16} - 192q^{22} + 624q^{23} + 952q^{25} - 2724q^{29} - 2792q^{37} - 632q^{43} - 2592q^{44} + 9792q^{46} - 2112q^{50} - 2076q^{53} + 672q^{58} + 2048q^{64} - 1488q^{65} - 29200q^{67} + 9696q^{71} + 1536q^{74} - 7948q^{79} + 1224q^{85} - 36480q^{86} - 1536q^{88} + 4992q^{92} + 6504q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.82843 0.707107
\(3\) 0 0
\(4\) 8.00000 0.500000
\(5\) 23.9515i 0.958062i 0.877798 + 0.479031i \(0.159012\pi\)
−0.877798 + 0.479031i \(0.840988\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 22.6274 0.353553
\(9\) 0 0
\(10\) 67.7452i 0.677452i
\(11\) −97.9706 −0.809674 −0.404837 0.914389i \(-0.632672\pi\)
−0.404837 + 0.914389i \(0.632672\pi\)
\(12\) 0 0
\(13\) − 104.211i − 0.616636i −0.951283 0.308318i \(-0.900234\pi\)
0.951283 0.308318i \(-0.0997661\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) 107.676i 0.372580i 0.982495 + 0.186290i \(0.0596464\pi\)
−0.982495 + 0.186290i \(0.940354\pi\)
\(18\) 0 0
\(19\) 38.3935i 0.106353i 0.998585 + 0.0531767i \(0.0169346\pi\)
−0.998585 + 0.0531767i \(0.983065\pi\)
\(20\) 191.612i 0.479031i
\(21\) 0 0
\(22\) −277.103 −0.572526
\(23\) 1021.50 1.93100 0.965500 0.260404i \(-0.0838558\pi\)
0.965500 + 0.260404i \(0.0838558\pi\)
\(24\) 0 0
\(25\) 51.3238 0.0821181
\(26\) − 294.755i − 0.436027i
\(27\) 0 0
\(28\) 0 0
\(29\) −621.603 −0.739124 −0.369562 0.929206i \(-0.620492\pi\)
−0.369562 + 0.929206i \(0.620492\pi\)
\(30\) 0 0
\(31\) 1519.14i 1.58079i 0.612600 + 0.790393i \(0.290124\pi\)
−0.612600 + 0.790393i \(0.709876\pi\)
\(32\) 181.019 0.176777
\(33\) 0 0
\(34\) 304.553i 0.263454i
\(35\) 0 0
\(36\) 0 0
\(37\) −562.235 −0.410691 −0.205345 0.978690i \(-0.565832\pi\)
−0.205345 + 0.978690i \(0.565832\pi\)
\(38\) 108.593i 0.0752031i
\(39\) 0 0
\(40\) 541.961i 0.338726i
\(41\) 1023.20i 0.608684i 0.952563 + 0.304342i \(0.0984365\pi\)
−0.952563 + 0.304342i \(0.901563\pi\)
\(42\) 0 0
\(43\) −3382.41 −1.82932 −0.914658 0.404228i \(-0.867540\pi\)
−0.914658 + 0.404228i \(0.867540\pi\)
\(44\) −783.765 −0.404837
\(45\) 0 0
\(46\) 2889.23 1.36542
\(47\) 3945.03i 1.78589i 0.450165 + 0.892945i \(0.351365\pi\)
−0.450165 + 0.892945i \(0.648635\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 145.166 0.0580663
\(51\) 0 0
\(52\) − 833.692i − 0.308318i
\(53\) −2190.60 −0.779851 −0.389925 0.920846i \(-0.627499\pi\)
−0.389925 + 0.920846i \(0.627499\pi\)
\(54\) 0 0
\(55\) − 2346.55i − 0.775718i
\(56\) 0 0
\(57\) 0 0
\(58\) −1758.16 −0.522639
\(59\) 2934.87i 0.843111i 0.906803 + 0.421556i \(0.138516\pi\)
−0.906803 + 0.421556i \(0.861484\pi\)
\(60\) 0 0
\(61\) 665.346i 0.178809i 0.995995 + 0.0894043i \(0.0284963\pi\)
−0.995995 + 0.0894043i \(0.971504\pi\)
\(62\) 4296.76i 1.11778i
\(63\) 0 0
\(64\) 512.000 0.125000
\(65\) 2496.03 0.590775
\(66\) 0 0
\(67\) −5925.38 −1.31998 −0.659989 0.751275i \(-0.729439\pi\)
−0.659989 + 0.751275i \(0.729439\pi\)
\(68\) 861.405i 0.186290i
\(69\) 0 0
\(70\) 0 0
\(71\) 4494.41 0.891571 0.445785 0.895140i \(-0.352924\pi\)
0.445785 + 0.895140i \(0.352924\pi\)
\(72\) 0 0
\(73\) − 8968.93i − 1.68304i −0.540225 0.841521i \(-0.681661\pi\)
0.540225 0.841521i \(-0.318339\pi\)
\(74\) −1590.24 −0.290402
\(75\) 0 0
\(76\) 307.148i 0.0531767i
\(77\) 0 0
\(78\) 0 0
\(79\) −10446.8 −1.67390 −0.836951 0.547277i \(-0.815664\pi\)
−0.836951 + 0.547277i \(0.815664\pi\)
\(80\) 1532.90i 0.239515i
\(81\) 0 0
\(82\) 2894.04i 0.430405i
\(83\) − 1269.28i − 0.184247i −0.995748 0.0921234i \(-0.970635\pi\)
0.995748 0.0921234i \(-0.0293654\pi\)
\(84\) 0 0
\(85\) −2579.00 −0.356954
\(86\) −9566.89 −1.29352
\(87\) 0 0
\(88\) −2216.82 −0.286263
\(89\) 3604.36i 0.455038i 0.973774 + 0.227519i \(0.0730614\pi\)
−0.973774 + 0.227519i \(0.926939\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8171.99 0.965500
\(93\) 0 0
\(94\) 11158.2i 1.26282i
\(95\) −919.584 −0.101893
\(96\) 0 0
\(97\) − 1950.53i − 0.207304i −0.994614 0.103652i \(-0.966947\pi\)
0.994614 0.103652i \(-0.0330529\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 410.590 0.0410590
\(101\) 11326.2i 1.11031i 0.831748 + 0.555153i \(0.187340\pi\)
−0.831748 + 0.555153i \(0.812660\pi\)
\(102\) 0 0
\(103\) − 16328.9i − 1.53915i −0.638555 0.769576i \(-0.720468\pi\)
0.638555 0.769576i \(-0.279532\pi\)
\(104\) − 2358.04i − 0.218014i
\(105\) 0 0
\(106\) −6195.95 −0.551438
\(107\) −4396.46 −0.384004 −0.192002 0.981395i \(-0.561498\pi\)
−0.192002 + 0.981395i \(0.561498\pi\)
\(108\) 0 0
\(109\) −14697.8 −1.23708 −0.618541 0.785753i \(-0.712276\pi\)
−0.618541 + 0.785753i \(0.712276\pi\)
\(110\) − 6637.03i − 0.548515i
\(111\) 0 0
\(112\) 0 0
\(113\) −2124.36 −0.166368 −0.0831842 0.996534i \(-0.526509\pi\)
−0.0831842 + 0.996534i \(0.526509\pi\)
\(114\) 0 0
\(115\) 24466.5i 1.85002i
\(116\) −4972.82 −0.369562
\(117\) 0 0
\(118\) 8301.07i 0.596170i
\(119\) 0 0
\(120\) 0 0
\(121\) −5042.77 −0.344428
\(122\) 1881.88i 0.126437i
\(123\) 0 0
\(124\) 12153.1i 0.790393i
\(125\) 16199.0i 1.03674i
\(126\) 0 0
\(127\) 8498.07 0.526881 0.263441 0.964676i \(-0.415143\pi\)
0.263441 + 0.964676i \(0.415143\pi\)
\(128\) 1448.15 0.0883883
\(129\) 0 0
\(130\) 7059.83 0.417741
\(131\) − 14472.3i − 0.843324i −0.906753 0.421662i \(-0.861447\pi\)
0.906753 0.421662i \(-0.138553\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −16759.5 −0.933366
\(135\) 0 0
\(136\) 2436.42i 0.131727i
\(137\) −12855.2 −0.684915 −0.342458 0.939533i \(-0.611259\pi\)
−0.342458 + 0.939533i \(0.611259\pi\)
\(138\) 0 0
\(139\) 5009.46i 0.259275i 0.991561 + 0.129638i \(0.0413814\pi\)
−0.991561 + 0.129638i \(0.958619\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12712.1 0.630436
\(143\) 10209.7i 0.499274i
\(144\) 0 0
\(145\) − 14888.3i − 0.708126i
\(146\) − 25368.0i − 1.19009i
\(147\) 0 0
\(148\) −4497.88 −0.205345
\(149\) −25398.9 −1.14404 −0.572022 0.820238i \(-0.693841\pi\)
−0.572022 + 0.820238i \(0.693841\pi\)
\(150\) 0 0
\(151\) −18308.8 −0.802981 −0.401491 0.915863i \(-0.631508\pi\)
−0.401491 + 0.915863i \(0.631508\pi\)
\(152\) 868.747i 0.0376016i
\(153\) 0 0
\(154\) 0 0
\(155\) −36385.6 −1.51449
\(156\) 0 0
\(157\) 35244.4i 1.42985i 0.699200 + 0.714926i \(0.253540\pi\)
−0.699200 + 0.714926i \(0.746460\pi\)
\(158\) −29548.1 −1.18363
\(159\) 0 0
\(160\) 4335.69i 0.169363i
\(161\) 0 0
\(162\) 0 0
\(163\) 145.939 0.00549281 0.00274641 0.999996i \(-0.499126\pi\)
0.00274641 + 0.999996i \(0.499126\pi\)
\(164\) 8185.58i 0.304342i
\(165\) 0 0
\(166\) − 3590.05i − 0.130282i
\(167\) − 31314.1i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(168\) 0 0
\(169\) 17701.0 0.619760
\(170\) −7294.50 −0.252405
\(171\) 0 0
\(172\) −27059.3 −0.914658
\(173\) 40115.8i 1.34037i 0.742196 + 0.670183i \(0.233784\pi\)
−0.742196 + 0.670183i \(0.766216\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6270.12 −0.202419
\(177\) 0 0
\(178\) 10194.7i 0.321760i
\(179\) 50154.5 1.56532 0.782661 0.622448i \(-0.213862\pi\)
0.782661 + 0.622448i \(0.213862\pi\)
\(180\) 0 0
\(181\) 63974.3i 1.95276i 0.216069 + 0.976378i \(0.430677\pi\)
−0.216069 + 0.976378i \(0.569323\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 23113.9 0.682711
\(185\) − 13466.4i − 0.393467i
\(186\) 0 0
\(187\) − 10549.0i − 0.301668i
\(188\) 31560.3i 0.892945i
\(189\) 0 0
\(190\) −2600.98 −0.0720492
\(191\) 42768.9 1.17236 0.586180 0.810181i \(-0.300631\pi\)
0.586180 + 0.810181i \(0.300631\pi\)
\(192\) 0 0
\(193\) −8772.58 −0.235512 −0.117756 0.993043i \(-0.537570\pi\)
−0.117756 + 0.993043i \(0.537570\pi\)
\(194\) − 5516.92i − 0.146586i
\(195\) 0 0
\(196\) 0 0
\(197\) 32825.0 0.845808 0.422904 0.906174i \(-0.361011\pi\)
0.422904 + 0.906174i \(0.361011\pi\)
\(198\) 0 0
\(199\) 26464.2i 0.668271i 0.942525 + 0.334136i \(0.108444\pi\)
−0.942525 + 0.334136i \(0.891556\pi\)
\(200\) 1161.33 0.0290331
\(201\) 0 0
\(202\) 32035.4i 0.785106i
\(203\) 0 0
\(204\) 0 0
\(205\) −24507.2 −0.583157
\(206\) − 46185.0i − 1.08834i
\(207\) 0 0
\(208\) − 6669.53i − 0.154159i
\(209\) − 3761.44i − 0.0861115i
\(210\) 0 0
\(211\) 67056.3 1.50617 0.753087 0.657921i \(-0.228564\pi\)
0.753087 + 0.657921i \(0.228564\pi\)
\(212\) −17524.8 −0.389925
\(213\) 0 0
\(214\) −12435.1 −0.271532
\(215\) − 81013.8i − 1.75260i
\(216\) 0 0
\(217\) 0 0
\(218\) −41571.6 −0.874749
\(219\) 0 0
\(220\) − 18772.4i − 0.387859i
\(221\) 11221.0 0.229746
\(222\) 0 0
\(223\) 56260.1i 1.13133i 0.824634 + 0.565666i \(0.191381\pi\)
−0.824634 + 0.565666i \(0.808619\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −6008.59 −0.117640
\(227\) 61148.1i 1.18667i 0.804954 + 0.593337i \(0.202190\pi\)
−0.804954 + 0.593337i \(0.797810\pi\)
\(228\) 0 0
\(229\) 33803.7i 0.644605i 0.946637 + 0.322303i \(0.104457\pi\)
−0.946637 + 0.322303i \(0.895543\pi\)
\(230\) 69201.6i 1.30816i
\(231\) 0 0
\(232\) −14065.3 −0.261320
\(233\) −8687.26 −0.160019 −0.0800094 0.996794i \(-0.525495\pi\)
−0.0800094 + 0.996794i \(0.525495\pi\)
\(234\) 0 0
\(235\) −94489.6 −1.71099
\(236\) 23479.0i 0.421556i
\(237\) 0 0
\(238\) 0 0
\(239\) 15715.0 0.275118 0.137559 0.990494i \(-0.456074\pi\)
0.137559 + 0.990494i \(0.456074\pi\)
\(240\) 0 0
\(241\) 5478.82i 0.0943307i 0.998887 + 0.0471654i \(0.0150188\pi\)
−0.998887 + 0.0471654i \(0.984981\pi\)
\(242\) −14263.1 −0.243547
\(243\) 0 0
\(244\) 5322.77i 0.0894043i
\(245\) 0 0
\(246\) 0 0
\(247\) 4001.05 0.0655813
\(248\) 34374.1i 0.558892i
\(249\) 0 0
\(250\) 45817.7i 0.733083i
\(251\) 24350.9i 0.386516i 0.981148 + 0.193258i \(0.0619054\pi\)
−0.981148 + 0.193258i \(0.938095\pi\)
\(252\) 0 0
\(253\) −100077. −1.56348
\(254\) 24036.2 0.372561
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) − 11606.7i − 0.175729i −0.996132 0.0878647i \(-0.971996\pi\)
0.996132 0.0878647i \(-0.0280043\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 19968.2 0.295388
\(261\) 0 0
\(262\) − 40933.8i − 0.596320i
\(263\) −120668. −1.74453 −0.872266 0.489031i \(-0.837351\pi\)
−0.872266 + 0.489031i \(0.837351\pi\)
\(264\) 0 0
\(265\) − 52468.3i − 0.747145i
\(266\) 0 0
\(267\) 0 0
\(268\) −47403.1 −0.659989
\(269\) − 27246.6i − 0.376537i −0.982118 0.188269i \(-0.939712\pi\)
0.982118 0.188269i \(-0.0602876\pi\)
\(270\) 0 0
\(271\) − 97043.3i − 1.32138i −0.750660 0.660689i \(-0.770264\pi\)
0.750660 0.660689i \(-0.229736\pi\)
\(272\) 6891.24i 0.0931450i
\(273\) 0 0
\(274\) −36359.9 −0.484308
\(275\) −5028.22 −0.0664889
\(276\) 0 0
\(277\) 29752.8 0.387765 0.193882 0.981025i \(-0.437892\pi\)
0.193882 + 0.981025i \(0.437892\pi\)
\(278\) 14168.9i 0.183335i
\(279\) 0 0
\(280\) 0 0
\(281\) 122275. 1.54855 0.774276 0.632849i \(-0.218114\pi\)
0.774276 + 0.632849i \(0.218114\pi\)
\(282\) 0 0
\(283\) 60700.7i 0.757916i 0.925414 + 0.378958i \(0.123718\pi\)
−0.925414 + 0.378958i \(0.876282\pi\)
\(284\) 35955.3 0.445785
\(285\) 0 0
\(286\) 28877.3i 0.353040i
\(287\) 0 0
\(288\) 0 0
\(289\) 71927.0 0.861184
\(290\) − 42110.6i − 0.500721i
\(291\) 0 0
\(292\) − 71751.4i − 0.841521i
\(293\) 29557.4i 0.344296i 0.985071 + 0.172148i \(0.0550707\pi\)
−0.985071 + 0.172148i \(0.944929\pi\)
\(294\) 0 0
\(295\) −70294.7 −0.807752
\(296\) −12721.9 −0.145201
\(297\) 0 0
\(298\) −71839.0 −0.808961
\(299\) − 106452.i − 1.19072i
\(300\) 0 0
\(301\) 0 0
\(302\) −51785.0 −0.567794
\(303\) 0 0
\(304\) 2457.19i 0.0265883i
\(305\) −15936.1 −0.171310
\(306\) 0 0
\(307\) − 107567.i − 1.14131i −0.821191 0.570654i \(-0.806690\pi\)
0.821191 0.570654i \(-0.193310\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −102914. −1.07091
\(311\) 43371.8i 0.448421i 0.974541 + 0.224211i \(0.0719804\pi\)
−0.974541 + 0.224211i \(0.928020\pi\)
\(312\) 0 0
\(313\) − 107746.i − 1.09980i −0.835230 0.549900i \(-0.814666\pi\)
0.835230 0.549900i \(-0.185334\pi\)
\(314\) 99686.3i 1.01106i
\(315\) 0 0
\(316\) −83574.6 −0.836951
\(317\) 155994. 1.55235 0.776174 0.630519i \(-0.217158\pi\)
0.776174 + 0.630519i \(0.217158\pi\)
\(318\) 0 0
\(319\) 60898.8 0.598449
\(320\) 12263.2i 0.119758i
\(321\) 0 0
\(322\) 0 0
\(323\) −4134.05 −0.0396251
\(324\) 0 0
\(325\) − 5348.53i − 0.0506370i
\(326\) 412.776 0.00388400
\(327\) 0 0
\(328\) 23152.3i 0.215202i
\(329\) 0 0
\(330\) 0 0
\(331\) 40218.7 0.367090 0.183545 0.983011i \(-0.441243\pi\)
0.183545 + 0.983011i \(0.441243\pi\)
\(332\) − 10154.2i − 0.0921234i
\(333\) 0 0
\(334\) − 88569.6i − 0.793947i
\(335\) − 141922.i − 1.26462i
\(336\) 0 0
\(337\) 90476.4 0.796665 0.398332 0.917241i \(-0.369589\pi\)
0.398332 + 0.917241i \(0.369589\pi\)
\(338\) 50065.9 0.438237
\(339\) 0 0
\(340\) −20632.0 −0.178477
\(341\) − 148831.i − 1.27992i
\(342\) 0 0
\(343\) 0 0
\(344\) −76535.1 −0.646761
\(345\) 0 0
\(346\) 113465.i 0.947782i
\(347\) −32074.5 −0.266379 −0.133190 0.991091i \(-0.542522\pi\)
−0.133190 + 0.991091i \(0.542522\pi\)
\(348\) 0 0
\(349\) − 116783.i − 0.958805i −0.877595 0.479402i \(-0.840853\pi\)
0.877595 0.479402i \(-0.159147\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −17734.6 −0.143132
\(353\) 124658.i 1.00039i 0.865912 + 0.500197i \(0.166739\pi\)
−0.865912 + 0.500197i \(0.833261\pi\)
\(354\) 0 0
\(355\) 107648.i 0.854180i
\(356\) 28834.8i 0.227519i
\(357\) 0 0
\(358\) 141858. 1.10685
\(359\) −99442.1 −0.771581 −0.385790 0.922587i \(-0.626071\pi\)
−0.385790 + 0.922587i \(0.626071\pi\)
\(360\) 0 0
\(361\) 128847. 0.988689
\(362\) 180947.i 1.38081i
\(363\) 0 0
\(364\) 0 0
\(365\) 214820. 1.61246
\(366\) 0 0
\(367\) − 88914.6i − 0.660147i −0.943955 0.330074i \(-0.892926\pi\)
0.943955 0.330074i \(-0.107074\pi\)
\(368\) 65375.9 0.482750
\(369\) 0 0
\(370\) − 38088.7i − 0.278223i
\(371\) 0 0
\(372\) 0 0
\(373\) −91352.9 −0.656606 −0.328303 0.944573i \(-0.606477\pi\)
−0.328303 + 0.944573i \(0.606477\pi\)
\(374\) − 29837.2i − 0.213312i
\(375\) 0 0
\(376\) 89265.9i 0.631408i
\(377\) 64778.2i 0.455770i
\(378\) 0 0
\(379\) 6328.51 0.0440578 0.0220289 0.999757i \(-0.492987\pi\)
0.0220289 + 0.999757i \(0.492987\pi\)
\(380\) −7356.68 −0.0509465
\(381\) 0 0
\(382\) 120969. 0.828984
\(383\) 13436.0i 0.0915949i 0.998951 + 0.0457974i \(0.0145829\pi\)
−0.998951 + 0.0457974i \(0.985417\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −24812.6 −0.166532
\(387\) 0 0
\(388\) − 15604.2i − 0.103652i
\(389\) −88296.3 −0.583503 −0.291752 0.956494i \(-0.594238\pi\)
−0.291752 + 0.956494i \(0.594238\pi\)
\(390\) 0 0
\(391\) 109990.i 0.719451i
\(392\) 0 0
\(393\) 0 0
\(394\) 92843.0 0.598077
\(395\) − 250218.i − 1.60370i
\(396\) 0 0
\(397\) 216587.i 1.37421i 0.726560 + 0.687103i \(0.241118\pi\)
−0.726560 + 0.687103i \(0.758882\pi\)
\(398\) 74852.1i 0.472539i
\(399\) 0 0
\(400\) 3284.72 0.0205295
\(401\) 5166.84 0.0321319 0.0160659 0.999871i \(-0.494886\pi\)
0.0160659 + 0.999871i \(0.494886\pi\)
\(402\) 0 0
\(403\) 158311. 0.974769
\(404\) 90609.9i 0.555153i
\(405\) 0 0
\(406\) 0 0
\(407\) 55082.5 0.332526
\(408\) 0 0
\(409\) − 38910.7i − 0.232607i −0.993214 0.116303i \(-0.962896\pi\)
0.993214 0.116303i \(-0.0371045\pi\)
\(410\) −69316.7 −0.412354
\(411\) 0 0
\(412\) − 130631.i − 0.769576i
\(413\) 0 0
\(414\) 0 0
\(415\) 30401.1 0.176520
\(416\) − 18864.3i − 0.109007i
\(417\) 0 0
\(418\) − 10639.0i − 0.0608900i
\(419\) − 263425.i − 1.50048i −0.661167 0.750239i \(-0.729938\pi\)
0.661167 0.750239i \(-0.270062\pi\)
\(420\) 0 0
\(421\) −16375.1 −0.0923889 −0.0461944 0.998932i \(-0.514709\pi\)
−0.0461944 + 0.998932i \(0.514709\pi\)
\(422\) 189664. 1.06503
\(423\) 0 0
\(424\) −49567.6 −0.275719
\(425\) 5526.32i 0.0305955i
\(426\) 0 0
\(427\) 0 0
\(428\) −35171.7 −0.192002
\(429\) 0 0
\(430\) − 229142.i − 1.23927i
\(431\) 250416. 1.34806 0.674028 0.738705i \(-0.264563\pi\)
0.674028 + 0.738705i \(0.264563\pi\)
\(432\) 0 0
\(433\) − 91495.1i − 0.488002i −0.969775 0.244001i \(-0.921540\pi\)
0.969775 0.244001i \(-0.0784601\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −117582. −0.618541
\(437\) 39219.0i 0.205368i
\(438\) 0 0
\(439\) 8859.37i 0.0459699i 0.999736 + 0.0229850i \(0.00731699\pi\)
−0.999736 + 0.0229850i \(0.992683\pi\)
\(440\) − 53096.3i − 0.274258i
\(441\) 0 0
\(442\) 31737.9 0.162455
\(443\) −300209. −1.52974 −0.764868 0.644187i \(-0.777196\pi\)
−0.764868 + 0.644187i \(0.777196\pi\)
\(444\) 0 0
\(445\) −86329.9 −0.435954
\(446\) 159127.i 0.799973i
\(447\) 0 0
\(448\) 0 0
\(449\) −116866. −0.579691 −0.289846 0.957073i \(-0.593604\pi\)
−0.289846 + 0.957073i \(0.593604\pi\)
\(450\) 0 0
\(451\) − 100243.i − 0.492836i
\(452\) −16994.9 −0.0831842
\(453\) 0 0
\(454\) 172953.i 0.839105i
\(455\) 0 0
\(456\) 0 0
\(457\) 73745.8 0.353106 0.176553 0.984291i \(-0.443505\pi\)
0.176553 + 0.984291i \(0.443505\pi\)
\(458\) 95611.4i 0.455805i
\(459\) 0 0
\(460\) 195732.i 0.925008i
\(461\) 88219.1i 0.415108i 0.978224 + 0.207554i \(0.0665502\pi\)
−0.978224 + 0.207554i \(0.933450\pi\)
\(462\) 0 0
\(463\) 297880. 1.38957 0.694784 0.719219i \(-0.255500\pi\)
0.694784 + 0.719219i \(0.255500\pi\)
\(464\) −39782.6 −0.184781
\(465\) 0 0
\(466\) −24571.3 −0.113150
\(467\) 187470.i 0.859601i 0.902924 + 0.429801i \(0.141416\pi\)
−0.902924 + 0.429801i \(0.858584\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −267257. −1.20985
\(471\) 0 0
\(472\) 66408.5i 0.298085i
\(473\) 331376. 1.48115
\(474\) 0 0
\(475\) 1970.50i 0.00873353i
\(476\) 0 0
\(477\) 0 0
\(478\) 44448.8 0.194538
\(479\) − 120489.i − 0.525142i −0.964913 0.262571i \(-0.915430\pi\)
0.964913 0.262571i \(-0.0845703\pi\)
\(480\) 0 0
\(481\) 58591.4i 0.253247i
\(482\) 15496.4i 0.0667019i
\(483\) 0 0
\(484\) −40342.2 −0.172214
\(485\) 46718.1 0.198610
\(486\) 0 0
\(487\) −267394. −1.12744 −0.563721 0.825966i \(-0.690631\pi\)
−0.563721 + 0.825966i \(0.690631\pi\)
\(488\) 15055.1i 0.0632184i
\(489\) 0 0
\(490\) 0 0
\(491\) −178364. −0.739851 −0.369926 0.929061i \(-0.620617\pi\)
−0.369926 + 0.929061i \(0.620617\pi\)
\(492\) 0 0
\(493\) − 66931.5i − 0.275383i
\(494\) 11316.7 0.0463730
\(495\) 0 0
\(496\) 97224.7i 0.395196i
\(497\) 0 0
\(498\) 0 0
\(499\) 263747. 1.05922 0.529610 0.848241i \(-0.322338\pi\)
0.529610 + 0.848241i \(0.322338\pi\)
\(500\) 129592.i 0.518368i
\(501\) 0 0
\(502\) 68874.7i 0.273308i
\(503\) − 480056.i − 1.89739i −0.316197 0.948694i \(-0.602406\pi\)
0.316197 0.948694i \(-0.397594\pi\)
\(504\) 0 0
\(505\) −271281. −1.06374
\(506\) −283060. −1.10555
\(507\) 0 0
\(508\) 67984.5 0.263441
\(509\) 60266.8i 0.232618i 0.993213 + 0.116309i \(0.0371062\pi\)
−0.993213 + 0.116309i \(0.962894\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11585.2 0.0441942
\(513\) 0 0
\(514\) − 32828.8i − 0.124259i
\(515\) 391101. 1.47460
\(516\) 0 0
\(517\) − 386497.i − 1.44599i
\(518\) 0 0
\(519\) 0 0
\(520\) 56478.6 0.208871
\(521\) 287316.i 1.05848i 0.848471 + 0.529242i \(0.177524\pi\)
−0.848471 + 0.529242i \(0.822476\pi\)
\(522\) 0 0
\(523\) − 276444.i − 1.01066i −0.862927 0.505328i \(-0.831371\pi\)
0.862927 0.505328i \(-0.168629\pi\)
\(524\) − 115778.i − 0.421662i
\(525\) 0 0
\(526\) −341300. −1.23357
\(527\) −163574. −0.588969
\(528\) 0 0
\(529\) 763619. 2.72876
\(530\) − 148403.i − 0.528311i
\(531\) 0 0
\(532\) 0 0
\(533\) 106629. 0.375336
\(534\) 0 0
\(535\) − 105302.i − 0.367900i
\(536\) −134076. −0.466683
\(537\) 0 0
\(538\) − 77065.1i − 0.266252i
\(539\) 0 0
\(540\) 0 0
\(541\) −59664.0 −0.203853 −0.101927 0.994792i \(-0.532501\pi\)
−0.101927 + 0.994792i \(0.532501\pi\)
\(542\) − 274480.i − 0.934355i
\(543\) 0 0
\(544\) 19491.4i 0.0658634i
\(545\) − 352034.i − 1.18520i
\(546\) 0 0
\(547\) 489585. 1.63627 0.818133 0.575030i \(-0.195010\pi\)
0.818133 + 0.575030i \(0.195010\pi\)
\(548\) −102841. −0.342458
\(549\) 0 0
\(550\) −14222.0 −0.0470147
\(551\) − 23865.5i − 0.0786082i
\(552\) 0 0
\(553\) 0 0
\(554\) 84153.6 0.274191
\(555\) 0 0
\(556\) 40075.7i 0.129638i
\(557\) −508534. −1.63912 −0.819558 0.572996i \(-0.805781\pi\)
−0.819558 + 0.572996i \(0.805781\pi\)
\(558\) 0 0
\(559\) 352486.i 1.12802i
\(560\) 0 0
\(561\) 0 0
\(562\) 345846. 1.09499
\(563\) 119817.i 0.378009i 0.981976 + 0.189004i \(0.0605260\pi\)
−0.981976 + 0.189004i \(0.939474\pi\)
\(564\) 0 0
\(565\) − 50881.7i − 0.159391i
\(566\) 171688.i 0.535927i
\(567\) 0 0
\(568\) 101697. 0.315218
\(569\) 97659.4 0.301640 0.150820 0.988561i \(-0.451809\pi\)
0.150820 + 0.988561i \(0.451809\pi\)
\(570\) 0 0
\(571\) 47651.4 0.146151 0.0730757 0.997326i \(-0.476719\pi\)
0.0730757 + 0.997326i \(0.476719\pi\)
\(572\) 81677.3i 0.249637i
\(573\) 0 0
\(574\) 0 0
\(575\) 52427.2 0.158570
\(576\) 0 0
\(577\) 131457.i 0.394849i 0.980318 + 0.197425i \(0.0632578\pi\)
−0.980318 + 0.197425i \(0.936742\pi\)
\(578\) 203440. 0.608949
\(579\) 0 0
\(580\) − 119107.i − 0.354063i
\(581\) 0 0
\(582\) 0 0
\(583\) 214614. 0.631425
\(584\) − 202944.i − 0.595045i
\(585\) 0 0
\(586\) 83601.0i 0.243454i
\(587\) 135493.i 0.393224i 0.980481 + 0.196612i \(0.0629940\pi\)
−0.980481 + 0.196612i \(0.937006\pi\)
\(588\) 0 0
\(589\) −58325.0 −0.168122
\(590\) −198823. −0.571167
\(591\) 0 0
\(592\) −35983.1 −0.102673
\(593\) − 188208.i − 0.535216i −0.963528 0.267608i \(-0.913767\pi\)
0.963528 0.267608i \(-0.0862332\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −203191. −0.572022
\(597\) 0 0
\(598\) − 301091.i − 0.841969i
\(599\) 78404.8 0.218519 0.109259 0.994013i \(-0.465152\pi\)
0.109259 + 0.994013i \(0.465152\pi\)
\(600\) 0 0
\(601\) − 254898.i − 0.705695i −0.935681 0.352848i \(-0.885213\pi\)
0.935681 0.352848i \(-0.114787\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −146470. −0.401491
\(605\) − 120782.i − 0.329983i
\(606\) 0 0
\(607\) 565684.i 1.53531i 0.640863 + 0.767655i \(0.278577\pi\)
−0.640863 + 0.767655i \(0.721423\pi\)
\(608\) 6949.97i 0.0188008i
\(609\) 0 0
\(610\) −45074.0 −0.121134
\(611\) 411118. 1.10124
\(612\) 0 0
\(613\) −414221. −1.10233 −0.551164 0.834397i \(-0.685816\pi\)
−0.551164 + 0.834397i \(0.685816\pi\)
\(614\) − 304246.i − 0.807026i
\(615\) 0 0
\(616\) 0 0
\(617\) −22896.0 −0.0601435 −0.0300717 0.999548i \(-0.509574\pi\)
−0.0300717 + 0.999548i \(0.509574\pi\)
\(618\) 0 0
\(619\) − 161155.i − 0.420593i −0.977638 0.210296i \(-0.932557\pi\)
0.977638 0.210296i \(-0.0674429\pi\)
\(620\) −291085. −0.757245
\(621\) 0 0
\(622\) 122674.i 0.317082i
\(623\) 0 0
\(624\) 0 0
\(625\) −355913. −0.911139
\(626\) − 304753.i − 0.777676i
\(627\) 0 0
\(628\) 281956.i 0.714926i
\(629\) − 60539.0i − 0.153015i
\(630\) 0 0
\(631\) 489038. 1.22824 0.614121 0.789212i \(-0.289511\pi\)
0.614121 + 0.789212i \(0.289511\pi\)
\(632\) −236385. −0.591814
\(633\) 0 0
\(634\) 441217. 1.09768
\(635\) 203542.i 0.504785i
\(636\) 0 0
\(637\) 0 0
\(638\) 172248. 0.423168
\(639\) 0 0
\(640\) 34685.5i 0.0846815i
\(641\) −119340. −0.290448 −0.145224 0.989399i \(-0.546390\pi\)
−0.145224 + 0.989399i \(0.546390\pi\)
\(642\) 0 0
\(643\) − 324224.i − 0.784194i −0.919924 0.392097i \(-0.871750\pi\)
0.919924 0.392097i \(-0.128250\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −11692.9 −0.0280192
\(647\) − 460781.i − 1.10074i −0.834920 0.550372i \(-0.814486\pi\)
0.834920 0.550372i \(-0.185514\pi\)
\(648\) 0 0
\(649\) − 287531.i − 0.682645i
\(650\) − 15127.9i − 0.0358057i
\(651\) 0 0
\(652\) 1167.51 0.00274641
\(653\) 559124. 1.31124 0.655620 0.755091i \(-0.272407\pi\)
0.655620 + 0.755091i \(0.272407\pi\)
\(654\) 0 0
\(655\) 346634. 0.807957
\(656\) 65484.6i 0.152171i
\(657\) 0 0
\(658\) 0 0
\(659\) 184372. 0.424546 0.212273 0.977210i \(-0.431913\pi\)
0.212273 + 0.977210i \(0.431913\pi\)
\(660\) 0 0
\(661\) − 79226.4i − 0.181329i −0.995881 0.0906645i \(-0.971101\pi\)
0.995881 0.0906645i \(-0.0288991\pi\)
\(662\) 113756. 0.259572
\(663\) 0 0
\(664\) − 28720.4i − 0.0651411i
\(665\) 0 0
\(666\) 0 0
\(667\) −634967. −1.42725
\(668\) − 250513.i − 0.561405i
\(669\) 0 0
\(670\) − 401416.i − 0.894222i
\(671\) − 65184.4i − 0.144777i
\(672\) 0 0
\(673\) −482896. −1.06616 −0.533082 0.846064i \(-0.678966\pi\)
−0.533082 + 0.846064i \(0.678966\pi\)
\(674\) 255906. 0.563327
\(675\) 0 0
\(676\) 141608. 0.309880
\(677\) 380329.i 0.829817i 0.909863 + 0.414908i \(0.136186\pi\)
−0.909863 + 0.414908i \(0.863814\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −58356.0 −0.126202
\(681\) 0 0
\(682\) − 420956.i − 0.905041i
\(683\) 342069. 0.733283 0.366642 0.930362i \(-0.380507\pi\)
0.366642 + 0.930362i \(0.380507\pi\)
\(684\) 0 0
\(685\) − 307901.i − 0.656191i
\(686\) 0 0
\(687\) 0 0
\(688\) −216474. −0.457329
\(689\) 228286.i 0.480884i
\(690\) 0 0
\(691\) 73062.6i 0.153017i 0.997069 + 0.0765084i \(0.0243772\pi\)
−0.997069 + 0.0765084i \(0.975623\pi\)
\(692\) 320927.i 0.670183i
\(693\) 0 0
\(694\) −90720.3 −0.188359
\(695\) −119984. −0.248402
\(696\) 0 0
\(697\) −110173. −0.226783
\(698\) − 330313.i − 0.677977i
\(699\) 0 0
\(700\) 0 0
\(701\) 288287. 0.586664 0.293332 0.956011i \(-0.405236\pi\)
0.293332 + 0.956011i \(0.405236\pi\)
\(702\) 0 0
\(703\) − 21586.2i − 0.0436783i
\(704\) −50160.9 −0.101209
\(705\) 0 0
\(706\) 352586.i 0.707386i
\(707\) 0 0
\(708\) 0 0
\(709\) −310512. −0.617711 −0.308856 0.951109i \(-0.599946\pi\)
−0.308856 + 0.951109i \(0.599946\pi\)
\(710\) 304475.i 0.603996i
\(711\) 0 0
\(712\) 81557.3i 0.160880i
\(713\) 1.55179e6i 3.05250i
\(714\) 0 0
\(715\) −244537. −0.478335
\(716\) 401236. 0.782661
\(717\) 0 0
\(718\) −281265. −0.545590
\(719\) − 10084.7i − 0.0195077i −0.999952 0.00975383i \(-0.996895\pi\)
0.999952 0.00975383i \(-0.00310479\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 364434. 0.699109
\(723\) 0 0
\(724\) 511794.i 0.976378i
\(725\) −31903.0 −0.0606954
\(726\) 0 0
\(727\) − 638552.i − 1.20817i −0.796921 0.604084i \(-0.793539\pi\)
0.796921 0.604084i \(-0.206461\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 607602. 1.14018
\(731\) − 364203.i − 0.681567i
\(732\) 0 0
\(733\) − 826084.i − 1.53750i −0.639547 0.768752i \(-0.720878\pi\)
0.639547 0.768752i \(-0.279122\pi\)
\(734\) − 251488.i − 0.466795i
\(735\) 0 0
\(736\) 184911. 0.341356
\(737\) 580513. 1.06875
\(738\) 0 0
\(739\) 769990. 1.40992 0.704962 0.709245i \(-0.250964\pi\)
0.704962 + 0.709245i \(0.250964\pi\)
\(740\) − 107731.i − 0.196733i
\(741\) 0 0
\(742\) 0 0
\(743\) 611777. 1.10819 0.554097 0.832452i \(-0.313064\pi\)
0.554097 + 0.832452i \(0.313064\pi\)
\(744\) 0 0
\(745\) − 608343.i − 1.09606i
\(746\) −258385. −0.464290
\(747\) 0 0
\(748\) − 84392.3i − 0.150834i
\(749\) 0 0
\(750\) 0 0
\(751\) 370622. 0.657129 0.328565 0.944481i \(-0.393435\pi\)
0.328565 + 0.944481i \(0.393435\pi\)
\(752\) 252482.i 0.446473i
\(753\) 0 0
\(754\) 183220.i 0.322278i
\(755\) − 438523.i − 0.769305i
\(756\) 0 0
\(757\) 298778. 0.521384 0.260692 0.965422i \(-0.416049\pi\)
0.260692 + 0.965422i \(0.416049\pi\)
\(758\) 17899.7 0.0311536
\(759\) 0 0
\(760\) −20807.8 −0.0360246
\(761\) − 803142.i − 1.38683i −0.720539 0.693414i \(-0.756106\pi\)
0.720539 0.693414i \(-0.243894\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 342151. 0.586180
\(765\) 0 0
\(766\) 38002.6i 0.0647674i
\(767\) 305847. 0.519893
\(768\) 0 0
\(769\) 452541.i 0.765253i 0.923903 + 0.382626i \(0.124980\pi\)
−0.923903 + 0.382626i \(0.875020\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −70180.7 −0.117756
\(773\) 1.04853e6i 1.75478i 0.479774 + 0.877392i \(0.340719\pi\)
−0.479774 + 0.877392i \(0.659281\pi\)
\(774\) 0 0
\(775\) 77967.8i 0.129811i
\(776\) − 44135.4i − 0.0732932i
\(777\) 0 0
\(778\) −249740. −0.412599
\(779\) −39284.2 −0.0647355
\(780\) 0 0
\(781\) −440320. −0.721882
\(782\) 311100.i 0.508729i
\(783\) 0 0
\(784\) 0 0
\(785\) −844159. −1.36989
\(786\) 0 0
\(787\) − 195967.i − 0.316398i −0.987407 0.158199i \(-0.949431\pi\)
0.987407 0.158199i \(-0.0505687\pi\)
\(788\) 262600. 0.422904
\(789\) 0 0
\(790\) − 707722.i − 1.13399i
\(791\) 0 0
\(792\) 0 0
\(793\) 69336.7 0.110260
\(794\) 612601.i 0.971710i
\(795\) 0 0
\(796\) 211714.i 0.334136i
\(797\) − 66453.8i − 0.104617i −0.998631 0.0523086i \(-0.983342\pi\)
0.998631 0.0523086i \(-0.0166579\pi\)
\(798\) 0 0
\(799\) −424784. −0.665387
\(800\) 9290.60 0.0145166
\(801\) 0 0
\(802\) 14614.0 0.0227207
\(803\) 878691.i 1.36272i
\(804\) 0 0
\(805\) 0 0
\(806\) 447772. 0.689266
\(807\) 0 0
\(808\) 256284.i 0.392553i
\(809\) −82304.6 −0.125755 −0.0628777 0.998021i \(-0.520028\pi\)
−0.0628777 + 0.998021i \(0.520028\pi\)
\(810\) 0 0
\(811\) 469097.i 0.713215i 0.934254 + 0.356608i \(0.116067\pi\)
−0.934254 + 0.356608i \(0.883933\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 155797. 0.235131
\(815\) 3495.45i 0.00526245i
\(816\) 0 0
\(817\) − 129863.i − 0.194554i
\(818\) − 110056.i − 0.164478i
\(819\) 0 0
\(820\) −196057. −0.291578
\(821\) 5379.52 0.00798099 0.00399050 0.999992i \(-0.498730\pi\)
0.00399050 + 0.999992i \(0.498730\pi\)
\(822\) 0 0
\(823\) −969806. −1.43181 −0.715905 0.698198i \(-0.753986\pi\)
−0.715905 + 0.698198i \(0.753986\pi\)
\(824\) − 369480.i − 0.544172i
\(825\) 0 0
\(826\) 0 0
\(827\) −838747. −1.22637 −0.613183 0.789941i \(-0.710111\pi\)
−0.613183 + 0.789941i \(0.710111\pi\)
\(828\) 0 0
\(829\) 883703.i 1.28587i 0.765920 + 0.642935i \(0.222284\pi\)
−0.765920 + 0.642935i \(0.777716\pi\)
\(830\) 85987.3 0.124818
\(831\) 0 0
\(832\) − 53356.3i − 0.0770795i
\(833\) 0 0
\(834\) 0 0
\(835\) 750020. 1.07572
\(836\) − 30091.5i − 0.0430558i
\(837\) 0 0
\(838\) − 745080.i − 1.06100i
\(839\) 1.12626e6i 1.59998i 0.600011 + 0.799992i \(0.295163\pi\)
−0.600011 + 0.799992i \(0.704837\pi\)
\(840\) 0 0
\(841\) −320891. −0.453696
\(842\) −46315.8 −0.0653288
\(843\) 0 0
\(844\) 536451. 0.753087
\(845\) 423965.i 0.593768i
\(846\) 0 0
\(847\) 0 0
\(848\) −140198. −0.194963
\(849\) 0 0
\(850\) 15630.8i 0.0216343i
\(851\) −574323. −0.793043
\(852\) 0 0
\(853\) 805486.i 1.10703i 0.832839 + 0.553516i \(0.186714\pi\)
−0.832839 + 0.553516i \(0.813286\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −99480.6 −0.135766
\(857\) − 980477.i − 1.33498i −0.744618 0.667491i \(-0.767368\pi\)
0.744618 0.667491i \(-0.232632\pi\)
\(858\) 0 0
\(859\) − 433780.i − 0.587873i −0.955825 0.293936i \(-0.905035\pi\)
0.955825 0.293936i \(-0.0949654\pi\)
\(860\) − 648111.i − 0.876299i
\(861\) 0 0
\(862\) 708284. 0.953220
\(863\) 303157. 0.407048 0.203524 0.979070i \(-0.434760\pi\)
0.203524 + 0.979070i \(0.434760\pi\)
\(864\) 0 0
\(865\) −960836. −1.28415
\(866\) − 258787.i − 0.345070i
\(867\) 0 0
\(868\) 0 0
\(869\) 1.02348e6 1.35532
\(870\) 0 0
\(871\) 617493.i 0.813946i
\(872\) −332573. −0.437375
\(873\) 0 0
\(874\) 110928.i 0.145217i
\(875\) 0 0
\(876\) 0 0
\(877\) 355000. 0.461562 0.230781 0.973006i \(-0.425872\pi\)
0.230781 + 0.973006i \(0.425872\pi\)
\(878\) 25058.1i 0.0325057i
\(879\) 0 0
\(880\) − 150179.i − 0.193929i
\(881\) 320002.i 0.412288i 0.978522 + 0.206144i \(0.0660915\pi\)
−0.978522 + 0.206144i \(0.933909\pi\)
\(882\) 0 0
\(883\) −1.08605e6 −1.39293 −0.696467 0.717589i \(-0.745246\pi\)
−0.696467 + 0.717589i \(0.745246\pi\)
\(884\) 89768.2 0.114873
\(885\) 0 0
\(886\) −849120. −1.08169
\(887\) − 1.28423e6i − 1.63229i −0.577848 0.816144i \(-0.696107\pi\)
0.577848 0.816144i \(-0.303893\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −244178. −0.308266
\(891\) 0 0
\(892\) 450080.i 0.565666i
\(893\) −151464. −0.189935
\(894\) 0 0
\(895\) 1.20128e6i 1.49967i
\(896\) 0 0
\(897\) 0 0
\(898\) −330548. −0.409904
\(899\) − 944299.i − 1.16840i
\(900\) 0 0
\(901\) − 235874.i − 0.290557i
\(902\) − 283531.i − 0.348487i
\(903\) 0 0
\(904\) −48068.7 −0.0588201
\(905\) −1.53228e6 −1.87086
\(906\) 0 0
\(907\) 48435.5 0.0588775 0.0294388 0.999567i \(-0.490628\pi\)
0.0294388 + 0.999567i \(0.490628\pi\)
\(908\) 489185.i 0.593337i
\(909\) 0 0
\(910\) 0 0
\(911\) −727675. −0.876801 −0.438400 0.898780i \(-0.644455\pi\)
−0.438400 + 0.898780i \(0.644455\pi\)
\(912\) 0 0
\(913\) 124352.i 0.149180i
\(914\) 208584. 0.249683
\(915\) 0 0
\(916\) 270430.i 0.322303i
\(917\) 0 0
\(918\) 0 0
\(919\) 87715.9 0.103860 0.0519299 0.998651i \(-0.483463\pi\)
0.0519299 + 0.998651i \(0.483463\pi\)
\(920\) 553613.i 0.654080i
\(921\) 0 0
\(922\) 249521.i 0.293526i
\(923\) − 468369.i − 0.549775i
\(924\) 0 0
\(925\) −28856.1 −0.0337251
\(926\) 842533. 0.982573
\(927\) 0 0
\(928\) −112522. −0.130660
\(929\) − 901913.i − 1.04504i −0.852627 0.522520i \(-0.824992\pi\)
0.852627 0.522520i \(-0.175008\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −69498.1 −0.0800094
\(933\) 0 0
\(934\) 530244.i 0.607830i
\(935\) 252666. 0.289017
\(936\) 0 0
\(937\) 1.24186e6i 1.41447i 0.706978 + 0.707236i \(0.250058\pi\)
−0.706978 + 0.707236i \(0.749942\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −755917. −0.855496
\(941\) 426245.i 0.481371i 0.970603 + 0.240685i \(0.0773722\pi\)
−0.970603 + 0.240685i \(0.922628\pi\)
\(942\) 0 0
\(943\) 1.04520e6i 1.17537i
\(944\) 187832.i 0.210778i
\(945\) 0 0
\(946\) 937274. 1.04733
\(947\) −27330.4 −0.0304752 −0.0152376 0.999884i \(-0.504850\pi\)
−0.0152376 + 0.999884i \(0.504850\pi\)
\(948\) 0 0
\(949\) −934665. −1.03782
\(950\) 5573.42i 0.00617554i
\(951\) 0 0
\(952\) 0 0
\(953\) −1.30093e6 −1.43242 −0.716208 0.697887i \(-0.754124\pi\)
−0.716208 + 0.697887i \(0.754124\pi\)
\(954\) 0 0
\(955\) 1.02438e6i 1.12319i
\(956\) 125720. 0.137559
\(957\) 0 0
\(958\) − 340794.i − 0.371331i
\(959\) 0 0
\(960\) 0 0
\(961\) −1.38425e6 −1.49888
\(962\) 165721.i 0.179072i
\(963\) 0 0
\(964\) 43830.6i 0.0471654i
\(965\) − 210117.i − 0.225635i
\(966\) 0 0
\(967\) 463079. 0.495225 0.247612 0.968859i \(-0.420354\pi\)
0.247612 + 0.968859i \(0.420354\pi\)
\(968\) −114105. −0.121774
\(969\) 0 0
\(970\) 132139. 0.140439
\(971\) − 63188.7i − 0.0670195i −0.999438 0.0335098i \(-0.989332\pi\)
0.999438 0.0335098i \(-0.0106685\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −756305. −0.797221
\(975\) 0 0
\(976\) 42582.2i 0.0447021i
\(977\) −1.43899e6 −1.50754 −0.753772 0.657136i \(-0.771768\pi\)
−0.753772 + 0.657136i \(0.771768\pi\)
\(978\) 0 0
\(979\) − 353121.i − 0.368432i
\(980\) 0 0
\(981\) 0 0
\(982\) −504490. −0.523154
\(983\) 400596.i 0.414572i 0.978280 + 0.207286i \(0.0664630\pi\)
−0.978280 + 0.207286i \(0.933537\pi\)
\(984\) 0 0
\(985\) 786209.i 0.810336i
\(986\) − 189311.i − 0.194725i
\(987\) 0 0
\(988\) 32008.4 0.0327906
\(989\) −3.45512e6 −3.53241
\(990\) 0 0
\(991\) 198549. 0.202172 0.101086 0.994878i \(-0.467768\pi\)
0.101086 + 0.994878i \(0.467768\pi\)
\(992\) 274993.i 0.279446i
\(993\) 0 0
\(994\) 0 0
\(995\) −633858. −0.640245
\(996\) 0 0
\(997\) − 914786.i − 0.920299i −0.887841 0.460150i \(-0.847796\pi\)
0.887841 0.460150i \(-0.152204\pi\)
\(998\) 745988. 0.748981
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 882.5.c.a.685.4 4
3.2 odd 2 294.5.c.a.97.2 4
7.2 even 3 126.5.n.b.73.1 4
7.3 odd 6 126.5.n.b.19.1 4
7.6 odd 2 inner 882.5.c.a.685.3 4
21.2 odd 6 42.5.g.a.31.2 yes 4
21.5 even 6 294.5.g.c.31.2 4
21.11 odd 6 294.5.g.c.19.2 4
21.17 even 6 42.5.g.a.19.2 4
21.20 even 2 294.5.c.a.97.1 4
84.23 even 6 336.5.bh.d.241.1 4
84.59 odd 6 336.5.bh.d.145.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.5.g.a.19.2 4 21.17 even 6
42.5.g.a.31.2 yes 4 21.2 odd 6
126.5.n.b.19.1 4 7.3 odd 6
126.5.n.b.73.1 4 7.2 even 3
294.5.c.a.97.1 4 21.20 even 2
294.5.c.a.97.2 4 3.2 odd 2
294.5.g.c.19.2 4 21.11 odd 6
294.5.g.c.31.2 4 21.5 even 6
336.5.bh.d.145.1 4 84.59 odd 6
336.5.bh.d.241.1 4 84.23 even 6
882.5.c.a.685.3 4 7.6 odd 2 inner
882.5.c.a.685.4 4 1.1 even 1 trivial